line segment

Definition Suppose $V$ is a vector space over $ℝ$ or $ℂ$, and $L$ is a subset of $V$. Then $L$ is a line segment if $L$ can be parametrized as

$$L=\{a+tb\mid t\in [0,1]\}$$

for some $a,b$ in $V$ with $b\ne 0$.

Sometimes one needs to distinguish between open and closed (http://planetmath.org/Closed) line segments. Then one defines a closed line segment as above, and an open line segment as a subset $L$ that can be parametrized as

$$L=\{a+tb\mid t\in (0,1)\}$$

for some $a,b$ in $V$ with $b\ne 0$.

If $x$ and $y$ are two vectors in $V$ and $x\ne y$, then we denote by $[x,y]$ the set connecting $x$ and $y$. This is , $\{\alpha x+(1-\alpha )y|0\le \alpha \le 1\}$. One can easily check that $[x,y]$ is a closed line segment.

Remarks

• An alternative, equivalent, definition is as follows: A (closed) line segment is a convex hull of two distinct points.

• A line segment is connected, non-empty set.

• If $V$ is a topological vector space, then a closed line segment is a closed set in $V$. However, an open line segment is an open set in $V$ if and only if $V$ is one-dimensional.

• More generally than above, the concept of a line segment can be defined in an ordered geometry.

Title	line segment
Canonical name	LineSegment
Date of creation	2013-03-22 14:19:01
Last modified on	2013-03-22 14:19:01
Owner	matte (1858)
Last modified by	matte (1858)
Numerical id	12
Author	matte (1858)
Entry type	Definition
Classification	msc 03-00
Classification	msc 51-00
Related topic	Interval
Related topic	LinearManifold
Related topic	LineInThePlane
Related topic	CircularSegment
Defines	open line segment
Defines	closed line segment